Abstract
For the non-Hermitian, PT-symmetric potentials V = m 2 x 2 + gx 2(ix)ν with ν = 1 and 2, we construct the Q operator which gives both the positive-definite metric and an equivalent Hermitian Hamiltonian h. For the case ν = 1, where the theory may be defined on the real axis, h is reasonable but complicated. For the case ν = 2, where the theory must initially be defined on a contour in the complex x plane, we first introduce a real parametrization of the contour, and then calculate Q and h as an expansion in an angle θ. Theresultant h has less desirable properties. However, Q is not uniquely determined, and it may be possible to exploit this ambiguity to produce a more acceptable equivalent Hamiltonian.
Similar content being viewed by others
References
C.M. Bender and S. Boettcher: Phys. Rev. Lett. 80 (1998) 5243.
C.M. Bender, D. Brody, and H.F. Jones: Phys. Rev. Lett. 89 (2002) 27040; 92 (2004) 119902 (E).
A. Mostafazadeh: J. Math. Phys. 43 (2002) 205; J. Phys. A 36 (2003) 7081.
C.M. Bender, D. Brody, and H.F. Jones: Phys. Rev. D 70 (2004) 025001.
H.F. Jones: J. Phys. A 38 (2005) 1741.
A. Mostafazadeh: quant-ph/0411137.
C.M. Bender, P.N. Meisinger, and Q. Wang: J. Phys. A 36 (2003) 1973. C.M. Bender and H.F. Jones: Phys. Lett. A 328 (2004) 102.
A. Mostafazadeh: J. Phys. A 38 (2005) 3213.
H.F. Jones and J. Mateo: work in progress.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Jones, H.F., Mateo, J. Pseudo-Hermitian Hamiltonians: tale of two potentials. Czech J Phys 55, 1117–1122 (2005). https://doi.org/10.1007/s10582-005-0116-9
Received:
Issue Date:
DOI: https://doi.org/10.1007/s10582-005-0116-9